scholarly journals Robust stochastic optimization with convex risk measures: A discretized subgradient scheme

2021 ◽  
Vol 17 (1) ◽  
pp. 81-99
Author(s):  
Haodong Yu ◽  
◽  
Jie Sun ◽  
◽  
Keyword(s):  
Stochastic Optimization ◽  
Risk Measures ◽  
Convex Risk Measures

scholarly journals Robust decision making and convex risk measures computation

2015 ◽  
Author(s):  
Γεώργιος Παπαγιάννης
Keyword(s):  
Decision Making ◽  
Probability Measure ◽  
Calculus Of Variations ◽  
Model Uncertainty ◽  
Risk Measures ◽  
Risk Measure ◽  
Probability Measures ◽  
Discrete Measure ◽  
Convex Risk Measures ◽  
Variance Risk

The main aim of the present thesis is to investigate the effect of diverging priors concerning model uncertainty on decision making. One of the main issues in the thesis is to assess the effect of different notions of distance in the space of probability measures and their use as loss functionals in the process of identifying the best suited model among a set of plausible priors. Another issue, is that of addressing the problem of ``inhomogeneous" sets of priors, i.e. sets of priors that highly divergent opinions may occur, and the need to robustly treat that case. As high degrees of inhomogeneity may lead to distrust of the decision maker to the priors it may be desirable to adopt a particular prior corresponding to the set which somehow minimizes the ``variability" among the models on the set. This leads to the notion of Frechet risk measure. Finally, an important problem is the actual calculation of robust risk measures. An account of their variational definition, the problem of calculation leads to the numerical treatment of problems of the calculus of variations for which reliable and effective algorithms are proposed. The contributions of the thesis are presented in the following three chapters. In Chapter 2, a statistical learning scheme is introduced for constructing the best model compatible with a set of priors provided by different information sources of varying reliability. As various priors may model well different aspects of the phenomenon the proposed scheme is a variational scheme based on the minimization of a weighted loss function in the space of probability measures which in certain cases is shown to be equivalent to weighted quantile averaging schemes. Therefore in contrast to approaches such as minimax decision theory in which a particular element of the prior set is chosen we construct for each prior set a probability measure which is not necessarily an element of it, a fact that as shown may lead to better description of the phenomenon in question. While treating this problem we also address the issue of the effect of the choice of distance functional in the space of measures on the problem of model selection. One of the key findings in this respect is that the class of Wasserstein distances seems to have the best performance as compared to other distances such as the KL-divergence. In Chapter 3, motivated by the results of Chapter 2, we treat the problem of specifying the risk measure for a particular loss when a set of highly divergent priors concerning the distribution of the loss is available. Starting from the principle that the ``variability" of opinions is not welcome, a fact for which a strong axiomatic framework is provided (see e.g. Klibanoff (2005) and references therein) we introduce the concept of Frechet risk measures, which corresponds to a minimal variance risk measure. Here we view a set of priors as a discrete measure on the space of probability measures and by variance we mean the variance of this discrete probability measure. This requires the use of the concept of Frechet mean. By different metrizations of the space of probability measures we define a variety of Frechet risk measures, the Wasserstein, the Hellinger and the weighted entropic risk measure, and illustrate their use and performance via an example related to the static hedging of derivatives under model uncertainty. In Chapter 4, we consider the problem of numerical calculation of convex risk measures applying techniques from the calculus of variations. Regularization schemes are proposed and the theoretical convergence of the algorithms is considered.

scholarly journals Robustness in the Optimization of Risk Measures

2021 ◽  
Author(s):  
Paul Embrechts ◽  
Alexander Schied ◽  
Ruodu Wang
Keyword(s):  
Value At Risk ◽  
Optimization Problem ◽  
Financial Regulation ◽  
Optimization Problems ◽  
Risk Measures ◽  
Risk Measurement ◽  
Loss Functions ◽  
Insurance Regulation ◽  
Convex Risk Measures ◽  
Comparative Advantages

We study issues of robustness in the context of Quantitative Risk Management and Optimization. We develop a general methodology for determining whether a given risk-measurement-related optimization problem is robust, which we call “robustness against optimization.” The new notion is studied for various classes of risk measures and expected utility and loss functions. Motivated by practical issues from financial regulation, special attention is given to the two most widely used risk measures in the industry, Value-at-Risk (VaR) and Expected Shortfall (ES). We establish that for a class of general optimization problems, VaR leads to nonrobust optimizers, whereas convex risk measures generally lead to robust ones. Our results offer extra insight on the ongoing discussion about the comparative advantages of VaR and ES in banking and insurance regulation. Our notion of robustness is conceptually different from the field of robust optimization, to which some interesting links are derived.

Weighted Scoring Rules and Convex Risk Measures

2022 ◽  
Author(s):  
Zachary J. Smith ◽  
J. Eric Bickel
Keyword(s):  
Risk Measures ◽  
Risk Measure ◽  
Mathematical Finance ◽  
Scoring Rules ◽  
Convex Risk Measures ◽  
University Of Texas ◽  
Probabilistic Forecasts ◽  
Expected Utility Maximization ◽  
The University ◽  
The Relationship

In Weighted Scoring Rules and Convex Risk Measures, Dr. Zachary J. Smith and Prof. J. Eric Bickel (both at the University of Texas at Austin) present a general connection between weighted proper scoring rules and investment decisions involving the minimization of a convex risk measure. Weighted scoring rules are quantitative tools for evaluating the accuracy of probabilistic forecasts relative to a baseline distribution. In their paper, the authors demonstrate that the relationship between convex risk measures and weighted scoring rules relates closely with previous economic characterizations of weighted scores based on expected utility maximization. As illustrative examples, the authors study two families of weighted scoring rules based on phi-divergences (generalizations of the Weighted Power and Weighted Pseudospherical Scoring rules) along with their corresponding risk measures. The paper will be of particular interest to the decision analysis and mathematical finance communities as well as those interested in the elicitation and evaluation of subjective probabilistic forecasts.

SET-VALUED LAW INVARIANT COHERENT AND CONVEX RISK MEASURES

2019 ◽  
Vol 22 (03) ◽  
pp. 1950004 ◽  
Author(s):  
YANHONG CHEN ◽  
YIJUN HU
Keyword(s):  
At Risk ◽  
Value At Risk ◽  
Risk Measures ◽  
Convex Risk Measures ◽  
New Class ◽  
The Relationship ◽  
Distortion Risk Measures

In this paper, we investigate representation results for set-valued law invariant coherent and convex risk measures, which can be considered as a set-valued extension of the multivariate scalar law invariant coherent and convex risk measures studied in the literature. We further introduce a new class of set-valued risk measures, named set-valued distortion risk measures, which can be considered as a set-valued version of multivariate scalar distortion risk measures introduced in the literature. The relationship between set-valued distortion risk measures and set-valued weighted value at risk is also given.

scholarly journals VALUATIONS AND DYNAMIC CONVEX RISK MEASURES

2007 ◽  
Vol 18 (1) ◽  
pp. 1-22 ◽  
Author(s):  
A. Jobert ◽  
L. C. G. Rogers
Keyword(s):  
Risk Measures ◽  
Convex Risk Measures
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